DiD Model with multiple interactions

[Hannah Pareik]

Hi!

I am running a DiD model with multiple interactions and want to plot them using the margins, dydx() command.
More specifically I am looking into how turnout changes after a payment is introduced in a specific state relative to all other states.

the normal DiD model (without additional interactions) looks like this:

 reg y i.post##i.treated i.year i.state education othercontrolvars, vce(cluster state)

Now I want to display the marginal effect of the treatment (a payment) on turnout at different levels of educational attainment.

 (A) reg y i.post##i.treated##c.education i.year i.state othercontrolvars, vce(cluster state)

To simplify this (and be able to use the margins command) I decided to generate a payment variable which is the interaction of post*treated=payment

 gen payment=post*treated
(B) reg y post treatyear i.payment##c.education i.year i.state othercontrolvars, vce(cluster state)
margins, dydx(payment) at(education=(0(1)7))
marginsplot

Running (A)- i.post##i.treated##c.education and running (B)- i.payment##c.education however leads to different results.
Can someone explain to me why this is the case?

Thank you!

Best
Hannah

[Andrew Musau]

gen payment=post*treated

A categorical by categorical interaction is not a single variable, even in the case where both variables are binary. For two 0/1 binary variables, the interaction leads to 4 variables:

0#0
0#1
1#0
1#1

A straight multiplication works for two continuous variables or a binary and continuous variable.

[Hannah Pareik]

Hi Andrew,

thank you for clarifying!

Do you know what I could to instead / do you know how I can run the margins, dydx command for the payment then?

The following command does not work so I was was wondering how I could implement this in order to plot the marginal effect of the payment by different levels of education.

margins, dydx(post#treated) at(education=(0(1)7))

Thank you!
Best
Hannah

[Andrew Musau]

margins, dydx(post#treated) at(education=(0(1)7))

You cannot have a marginal effect for an interaction because it is not possible to change the interaction by one unit holding constant all other variables, where all other variables include the variables used to create the interaction. Therefore, there is no notion of a marginal effect for an interaction term. In case you want to disassociate the interaction terms from the variables defining the interaction, as implied in #1, you can do so specifying the full set of interactions and replicating the underlying regression. Here is how.

webuse nlswork, clear
set seed 03112022
keep if idcode<10
gen treat = idcode > 5
gen byte post = year >= 80
gen education = runiformint(1,7)
*SIMILAR TO YOUR REGRESSION
reg y i.post##i.treat##c.education i.idcode
*FULL SET OF INTERACTIONS
gen pay00= 0.post#0.treat
gen pay01= 0.post#1.treat
gen pay10= 1.post#0.treat
gen pay11= 1.post#1.treat
*RECREATE REGRESSION
regress y post treat education c.pay01#c.treat  c.pay01#c.education c.pay10#c.treat  c.pay10#c.education c.pay11#c.treat c.pay11#c.education i.idcode
*MARGINS
margins, dydx(pay??) at(education=(1/7))
marginsplot, recast(line) scheme(s1color)

Res.:

. *SIMILAR TO YOUR REGRESSION

. 
. reg y i.post##i.treat##c.education i.idcode
note: 9.idcode omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =        97
-------------+----------------------------------   F(13, 83)       =     25.41
       Model |  3131.91474        13  240.916519   Prob > F        =    0.0000
    Residual |  786.971856        83  9.48158863   R-squared       =    0.7992
-------------+----------------------------------   Adj R-squared   =    0.7677
       Total |   3918.8866        96  40.8217354   Root MSE        =    3.0792

----------------------------------------------------------------------------------------
                  year |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------------+----------------------------------------------------------------
                1.post |   11.15809   2.011975     5.55   0.000     7.156349    15.15982
               1.treat |  -2.152177   1.994364    -1.08   0.284    -6.118886    1.814532
                       |
            post#treat |
                  1 1  |    2.69253   3.367028     0.80   0.426    -4.004353    9.389413
                       |
             education |   .2272526   .2852377     0.80   0.428    -.3400737    .7945789
                       |
      post#c.education |
                    1  |  -.1266925   .4573676    -0.28   0.782    -1.036378    .7829933
                       |
     treat#c.education |
                    1  |   .2858844   .4266369     0.67   0.505    -.5626792    1.134448
                       |
post#treat#c.education |
                  1 1  |   -.315252   .7020794    -0.45   0.655     -1.71166    1.081156
                       |
                idcode |
                    2  |  -.0686068   1.279731    -0.05   0.957     -2.61394    2.476727
                    3  |  -1.143307   1.238566    -0.92   0.359    -3.606765    1.320151
                    4  |  -1.190976   1.352399    -0.88   0.381    -3.880843    1.498892
                    5  |  -1.984296   1.368529    -1.45   0.151    -4.706244    .7376534
                    6  |  -.7973263   1.179724    -0.68   0.501    -3.143749    1.549096
                    7  |  -1.481915   1.413745    -1.05   0.298    -4.293797    1.329967
                    9  |          0  (omitted)
                       |
                 _cons |   73.14695   1.249922    58.52   0.000     70.66091      75.633
----------------------------------------------------------------------------------------

. 
. *RECREATE REGRESSION

. 
. regress y post treat education c.pay01#c.treat  c.pay01#c.education c.pay10#c.treat  c.pay10#c.education c.pay11#c.treat c.pay11#c
> .education i.idcode
note: c.pay10#c.treat omitted because of collinearity
note: c.pay11#c.treat omitted because of collinearity
note: 9.idcode omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =        97
-------------+----------------------------------   F(13, 83)       =     25.41
       Model |  3131.91474        13  240.916519   Prob > F        =    0.0000
    Residual |  786.971856        83  9.48158863   R-squared       =    0.7992
-------------+----------------------------------   Adj R-squared   =    0.7677
       Total |   3918.8866        96  40.8217354   Root MSE        =    3.0792

-------------------------------------------------------------------------------------
               year |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------------+----------------------------------------------------------------
               post |   11.15809   2.011975     5.55   0.000     7.156349    15.15982
              treat |   .5403531    2.96346     0.18   0.856    -5.353849    6.434555
          education |   .2272526   .2852377     0.80   0.428    -.3400737    .7945789
                    |
    c.pay01#c.treat |   -2.69253   3.367028    -0.80   0.426    -9.389413    4.004353
                    |
c.pay01#c.education |   .2858844   .4266369     0.67   0.505    -.5626792    1.134448
                    |
    c.pay10#c.treat |          0  (omitted)
                    |
c.pay10#c.education |  -.1266925   .4573676    -0.28   0.782    -1.036378    .7829933
                    |
    c.pay11#c.treat |          0  (omitted)
                    |
c.pay11#c.education |  -.1560601   .5138954    -0.30   0.762    -1.178177    .8660572
                    |
             idcode |
                 2  |  -.0686068   1.279731    -0.05   0.957     -2.61394    2.476727
                 3  |  -1.143307   1.238566    -0.92   0.359    -3.606765    1.320151
                 4  |  -1.190976   1.352399    -0.88   0.381    -3.880843    1.498892
                 5  |  -1.984296   1.368529    -1.45   0.151    -4.706244    .7376534
                 6  |  -.7973263   1.179724    -0.68   0.501    -3.143749    1.549096
                 7  |  -1.481915   1.413745    -1.05   0.298    -4.293797    1.329967
                 9  |          0  (omitted)
                    |
              _cons |   73.14695   1.249922    58.52   0.000     70.66091      75.633
-------------------------------------------------------------------------------------